Here we are giving the details on how to calculate the greatest common factor of two polynomials using the long division method. Students must have a look at the complete procedure to find the H.C.F of Polynomials by Long Division Method and few solved examples on it. We are using this long division method especially when there is no scope to find the highest common factor of polynomials by factorization.
How to find H.C.F of Polynomials by Long Division?
The step by step procedure to evaluate the G.C.F of two polynomials by long division method manually is provided. For the sake of your comfort and convenience, the steps are listed below make use of them.
- Arrange the given two polynomials in the descending order of powers of any of its variables.
- If there is any common factor in the polynomials, then separate it. Multiply the common factors to get the H.C.F of them.
- Just like the determination of G.C.F by the method of division in arithmetic, here also division is not complete. In every step, the division of that step is divided by the obtained remainder. At any step, if any common factor is available in the remainder then it should be taken out, then perform division.
- In every step, the term in the quotient should be found by comparing the first term of the dividend with the first term of the divisor. Sometimes, if necessary, the dividend may be multiplied by a multiplier of a factor.
Highest Common Factor of Polynomials by Long Division Method
Example 1.
Find the H.C.F of 6x³ – 17x² – 5x + 6, 6x³ – 5x² – 3x + 2 and 3x³ – 7x² + 4 by using the long division method.
Solution:
Given three polynomials are 6x³ – 17x² – 5x + 6, 6x³ – 5x² – 3x + 2 and 3x³ – 7x² + 4
All the polynomials are arranged in the descending order of the powers of the variable ‘x’. And the polynomials have no common factors between them. So, by the long division method
The H.C.F. of 6x³ – 17x² – 5x + 6, 6x³ – 5x² – 3x + 2 is 6x² + x – 2.
Now, it is to be seen whether the third expression 3x³ – 7x² + 4 is divisible by 6x² + x – 2 or not. If it is not, then the H.C.F. of them is to be determined by the division method.
Therefore, the H.C.F. of 6x³ – 17x² – 5x + 6, 6x³ – 5x² – 3x + 2 and 3x³ – 7x² + 4 is 3x + 2.
Example 2.
Calculate the Highest Common Factor of polynomials 7a³ – 28a² + 49a – 42, 5a³ – 25a² + 50a – 40 by using the long division method?
Solution:
Given two polynomials are 7a³ – 28a² + 49a – 42, 5a³ – 25a² + 50a – 40
Consider f(a) = 7a³ – 28a² + 49a – 42, g(a) = 5a³ – 25a² + 50a – 40
Take out the common factors,
f(a) = 7(a³ – 4a² + 7a – 6)
g(a) = 5(a³ – 5a² + 10a – 8)
At the time of writing the final result the H.C.F. of 7 and 5 i.e. 35 is to be multiplied with the divisor of the last step.
Therefore, the H.C.F of 7a³ – 28a² + 49a – 42, 5a³ – 25a² + 50a – 40 is 35 * 2(a – 2) = 70(a – 2).
Example 3.
Find the G.C.F of x⁶ + 2x⁵ + 5x³ + 4x² + 6 and x³ + 2 by using the long division method?
Solution:
Given two polynomials are x⁶ + 2x⁵ + 5x³ + 4x² + 6 and x³ + 2.
By arranging the two polynomials in the descending order of powers of x we get,
x⁶ + 2x⁵ + 0x⁴ + 5x³ + 4x² + 0x + 6 and x³ + 0x² + 0x + 2
There are no common factors in those polynomials. So, dive them by using the long division method
Therefore, the H.C.F. of x⁶ + 2x⁵ + 5x³ + 4x² + 6 and x³ + 2 is x³ + 2x² + 3.